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Ancient Astronomy
Dr. Jay Maron

Aristarchus

In Ancient Greece, Aristarchus (310 BCE - 230 BCE) used a lunar eclipse to
measure the size of the moon relative to the size of the Earth.  He found that

Size of moon  ~  1/3  Size of Earth

Using this plus Eratosthenes' measurement of the size of the Earth, Aristarchus
calculated that the distance to the moon is ~ 60 Earth radii.

Scale model of the solar system

In the following diagrams, all sizes and distances to scale, unless an object
is too small to be visible, in which case a green dot is used.

Distance to the sun

Aristarchus measured the distance to the sun by observing the moon when it is
half illuminated.  In this configuration, the Earth-Moon-Sun system is at a
right angle and one measures the Moon-Earth-Sun angle.

If the sun were infinitely far away the angel would be 90 degrees,
and if the Sun is at a finite distance the angle is less than 90 degrees.

Aristarchus was unable to measure this angle precisely, but he was able to
show that it is at least 87 degrees.  This established that the sun is
at least 20 times more distant than the moon.

Size of the sun

A solar eclipse tells us that the sun and moon have the same angular size.

Sun diameter  =  Moon diameter  *  Distance to sun  /  Distance to moon

Using this, Aristarchus estalished that the minimum size for the sun is

Sun diameter  >  1/3 Earth diameters  *  20
>  7  Earth diameters

This estalished that the sun is larger than the Earth, and was the first solid
clue for heliocentrism.

The measurement of the distance to the sun was refined by Aryabhata (476 CE -
550 CE).

Map of the world

Eratosthenes' map of the world
Ptolemy's map of the world

Ptolemy developed a system of latitude and longitude for mapping the world.
His map covered 1/4 of the globe and was the standard until the Renaissance.

Timeline

-310  -230  Aristarchus.  Measures the distance to the sun.
-276  -195  Eratosthenes
-240        Eratosthenes measures the Earth's circumference to 20% error.
90   168  Ptolemy
476   550  Aryabhata.  Refines the measurement of the distance to the sun.
1473  1543  Copernicus
1546  1601  Brahe
1600        Brahe measures accurate positions for the stars and planets over a period of 10 years.
1608        Lippershey invents the telescope
1609        Galileo builds a telescope
1609        Kepler uses Brahe's data to establish that planets orbit in ellipses.
1610        Galileo discovers the moons of Jupiter.
1639        Horrocks measures the transit of Venus, producing a value for the
distance to the sun that was low by a factor of 1/2.
1571  1630  Kepler
1564  1642  Galileo
1643  1727  Newton
1670        Picard measures the size of the Earth accurate to 1%.
1672        Richter and Cassini measure the parallax of Mars, which yielded a
value for the distance to the sun accurate to 10%.
1676        Romer uses the moons of Jupiter to measure the speed of light.
1729        Bradley measures the deflection of starlight due to the Earth's motion,
which gives a measurement for V/C, where V is the Earth's velocity.
1849        Fizeau makes the first measurement of the speed of light that doesn't
require astronomical data.  5% error.
1862        Focault measures the speed of light to .2% error.
1863        First measurements of stellar parallax and the distance to nearby stars.
2003        WMAP mission measures the Hubble constant to 5% precision.
Previous to this, the Hubble constant had an error of ~ 20%.

R  =  Earth-sun distance
V  =  Earth orbital velocity
T  =  Earth orbital time (1 year)
=  2 Pi R / V
t  =  Time for light to cross the Earth's orbit
=  2 R / C
C  =  Speed of light

Brahe's data consisted of measurements of angles between different objects.
This data could be used to establish the shape of orbits but not their size.
For example, if the size of the solar system were doubled along with the speeds of the
planets, the angles would stay the same and you wouldn't be able to tell the difference.

In 1639, Horrocks used a transit of Venus to measure the distance to the sun,
but this method is incapable of giving an accurate value, and it can only be
done once per century.

In 1672, Richter and Cassini measured the parallax of Mars which gives a result
for the distance to the sun that is more accurate than the Venus method.  The
Mars method has an advantage over the Venus method in that it can be done once
every 26 months, when Mars is at closest approach.

In 1676, Romer used the moons of Jupiter to measure the time it takes for light to
cross the Earth's orbit.  This gives a value for R/C.

In 1729, Bradley measured the deflection of starlight due to the Earth's motion,
which gives a measurement of V/C, or equivalently, a measurement of R/C.

In 1849, Fizeau produced the first measurement for the speed of light that was
independent of the Earth-sun distance R.

According to the Hubble law, distant galaxies recede from us with a speed of

Speed of galaxy  =  Hubble constant  *  Distance

Speed is easy to measure (from the redshift) and distance is difficult to measure.
Previous to 2003, the value of the Hubble constant was not well determined.

Parallax

Resolution of human vision        60    arcseconds
Resolution of a 10 cm telescope    1.0  arcseconds
Parallax of Alpha Centauri         1.4  arcseconds       Nearest star
Parallax of 61 Cygni                .31 arcseconds

Since the parallax of stars is undetectable to human vision, an astronomer in
Ancient Greece could conclude that the stars are vastly more distant than the sun.

First parallax measurements were done in 1838, when Bessel measured the parallax
of 61 Cygni, and Struve & Henderson measured the parallax of Alpha Centauri.

The limit of telescope resolution is 1 arcsecond due to air turbulence.

Solar system data

Geosynchronus orbit  =     6.6    Earth radii
Distance to sun      = 23500      Earth radii
Moon orbital radius  =    .00257  A.U.           (Astronomical unit = Earth orbit radius)
Distance to Earth L2 =    .01     A.U.
Moon orbital radius  =    .257    Earth Hill radii    (Distance to Earth L2)
Earth-Venus distance =  28        Earth Hill radii
Earth-Mars distance  =  52        Earth Hill radii
Earth-sun  distance  = 100        Earth Hill radii

1 Earth radius  =  6371 km
1 A.U.          =  1.50*10^11 meters
1 Earth mass    =  5.974*10^24 kg

The distance to the Lagrange point L2 is the "Hill radius", an indicator of a
planet's gravitational influence.

If a moon is outside of 1/3 Hill radii, it will be stolen by the sun.
The moon is barely within this distance.

If two planets are within ~ 10 Hill radii of each other, their gravity will disrupt
each other's orbits.

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